How could I solve this chain rule problem?To find: d d t [ f ( c ( t ) ) ] | t = 0 Where c ( t ) is such that d d t c ( t ) = F ( c ( t ) ), with F = ∇ f, c ( 0 ) = ( − π 2 , π 2 ) and f ( x , y ) = sin ⁡ ( x + y ) x yI think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?Thanks in advance

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How could I solve this chain rule problem?To find:       d          d      t        [  f  (      c    (  t  )  )  ]      |          t      =      0      Where       c    (  t  ) is such that       d          d      t            c    (  t  )  =      F    (  c  (  t  )  ), with       F    =  ∇  f,       c    (  0  )  =  (  −      π    2    ,      π    2    ) and   f  (  x  ,  y  )  =  sin  ⁡  (  x  +  y  )  x  yI think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?Thanks in advance

iskakanjulc · Accepted Answer

Let   g  =  f  ∘  c, i.e. let  g  (  t  )  =  f  (  c  (  t  )  )  =  f  (      c    1    (  t  )  ,      c    2    (  t  )  )  .Then by the chain rule      g    ′    (  t  )  =      f    x    (  c  (  t  )  )      c    1    ′    (  t  )  +      f    y    (  c  (  t  )  )      c    2    ′    (  t  )  =  ∇  f  (  c  (  t  )  )  ⋅      c    ′    (  t  )  =  ∇  f  (  c  (  t  )  )  ⋅  ∇  f  (  c  (  t  )  )  ,where the second equality follows from the assumption about       c    ′    (  t  )Since      f    x    (  x  ,  y  )  =  y  [  sin  ⁡  (  x  +  y  )  +  x  cos  ⁡  (  x  +  y  )  ]and      f    y    (  x  ,  y  )  =  x  [  sin  ⁡  (  x  +  y  )  +  y  cos  ⁡  (  x  +  y  )  ]we have      f    x    (  c  (  0  )  )  =      f    x        (    −          π      2        ,          π      2        )    =  −            π      2        4        f    y    (  c  (  0  )  )  =      f    y        (    −          π      2        ,          π      2        )    =  −            π      2        4  or  ∇  f  (  c  (  0  )  )  =      (    −                  π        2            4        ,    −                  π        2            4        )    .Thus      g    ′    (  0  )  =  ∇  f  (  c  (  0  )  )  ⋅  ∇  f  (  c  (  0  )  )  =      (    −                  π        2            4        ,    −                  π        2            4        )    ⋅      (    −                  π        2            4        ,    −                  π        2            4        )    =            π      4        8    .

How could I solve this chain rule problem? To find: d d t </mro

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